I forgot what the quick and easy definition of "linear combination was. I seem to remember something about "closed under addition and scalar multiplication". Perhaps somebody could help me re-learn this concept. I took Calc III and Linear Algebra and passed with flying colors but time has eaten my brain.

A linear combination of, say, x1, x2, ..., xn must involve only multiplication by numbers and addition or subtraction: a1x1+ a2x2+ ...+ anxn where every "a" is a number. Anything more complicated than just "multiply by numbers and add", for example x2 or x/y or cos(x) is NOT linear.

A linear operator is an operator that "preserves" the operations: T(ax+ by)= aT(x)+ bT(y) where a and b are numbers. Matrix multiplication of vectors, differentiation of functions and integration of functions are examples of linear operators.